Problem: Let $S$ be the set of positive real numbers.  Let $f : S \to \mathbb{R}$ be a function such that
\[f(x) f(y) = f(xy) + 2005 \left( \frac{1}{x} + \frac{1}{y} + 2004 \right)\]for all $x,$ $y > 0.$

Let $n$ be the number of possible values of $f(2),$ and let $s$ be the sum of all possible values of $f(2).$  Find $n \times s.$
Explanation: Setting $y = 1,$ we get
\[f(x) f(1) = f(x) + \frac{2005}{x} + 2005^2.\]The value $f(1)$ cannot be 1, and so we can solve for $f(x)$ to get
\[f(x) = \frac{2005/x + 2005^2}{f(1) - 1}.\]In particular,
\[f(1) = \frac{2005 + 2005^2}{f(1) - 1}.\]Then $f(1)^2 - f(1) - 2005^2 - 2005 = 0,$ which factors as $(f(1) - 2006)(f(1) + 2005) = 0.$  Hence, $f(1) = 2006$ or $f(1) = -2005.$

If $f(1) = 2006,$ then
\[f(x) = \frac{2005/x + 2005^2}{2005} = \frac{1}{x} + 2005.\]We can check that this function works.

If $f(1) = -2005,$ then
\[f(x) = \frac{2005/x + 2005^2}{-2006}.\]We can check that this function does not work.

Therefore,
\[f(x) = \frac{1}{x} + 2005,\]so $n = 1$ and $s = \frac{1}{2} + 2005 = \frac{4011}{2},$ so $n \times s = \boxed{\frac{4011}{2}}.$